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[[_TOC_]]
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# Introduction
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W-SLDA Toolkit utilizes a local pairing field $`\Delta(\bm{r})`$. In such a case, renormalization
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W-SLDA Toolkit utilizes a local pairing field $`\Delta(\bm{r})`$. In such a case, a renormalization
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procedure is required. There are two predefined regularization schemes that can be selected in `predefines.h` file:
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```c
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/**
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| ... | ... | @@ -16,7 +16,7 @@ procedure is required. There are two predefined regularization schemes that can |
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```c
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#define REGULARIZATION_SCHEME SPHERICAL_CUTOFF
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```
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The effective coupling constant is computed according to prescription:
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The effective coupling constant is computed according to the prescription:
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```math
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\dfrac{1}{g_{\textrm{eff}}}=\dfrac{1}{g_0} - \dfrac{m}{2\alpha_+}\dfrac{k_c}{\hbar^2\pi^2}
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\left(
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| ... | ... | @@ -35,11 +35,11 @@ where |
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&= E_c.
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\end{aligned}
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```
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with $`\mu_{+} = (\mu_a - V_a + \mu_b - V_b)/2`$ being average local chemical potential and $`\frac{m}{2\alpha_+}=m_r`$ is reduced mass. $`E_c`$ stands for energy cutoff scale that can be controlled by tag:
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with $`\mu_{+} = (\mu_a - V_a + \mu_b - V_b)/2`$ being average local chemical potential and $`\frac{m}{2\alpha_+}=m_r`$ is reduced mass. $`E_c`$ stands for the energy cutoff scale that can be controlled by the tag:
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```bash
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# ec 4.9348022 # energy cut-off for regularization scheme, default ec = 0.5*(pi/DX)^2
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```
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For more info see [arXiv:1008.3933](https://arxiv.org/abs/1008.3933).
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For more info, see [arXiv:1008.3933](https://arxiv.org/abs/1008.3933).
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*Note*: the spherical cut-off scheme results in a significant decrease in memory consumption and improved performance of `td` codes.
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# Renormalization with cubic cutoff
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| ... | ... | @@ -47,45 +47,41 @@ For more info see [arXiv:1008.3933](https://arxiv.org/abs/1008.3933). |
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```c
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#define REGULARIZATION_SCHEME CUBIC_CUTOFF
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```
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The effective coupling constant is computed according to prescription:
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The effective coupling constant is computed according to the prescription:
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```math
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\dfrac{1}{g_{\textrm{eff}}}=\dfrac{1}{g_0} - \dfrac{m}{2\alpha_+}\dfrac{K}{2\hbar^2\pi^2 dx},
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```
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where $`K=2.442 75`$ is a numerical constant. In this formula, we assume that all states contribute to the densities. Physically it means that we take into account states up to the maximal value of energy set by lattice, which is of the order $`E_c\approx 3\frac{\hbar^2\pi^2}{2mdx^2}`$ (assuming that $`dx=dy=dz`$).
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where $`K=2.442 75`$ is a numerical constant. In this formula, we assume that all states contribute to the densities. Physically, it means that we take into account states up to the maximal value of energy set by the lattice, which is of the order $`E_c\approx 3\frac{\hbar^2\pi^2}{2mdx^2}`$ (assuming that $`dx=dy=dz`$).
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*Note*: when working with this renormalization scheme value of tag `ec` will be ignored.
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# Custom renormalization scheme
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Static codes allow for defining your own renormalization scheme. You need to provide the formula in `void modify_potentials(...)` function. See [here](Strict 2D or 1D mode) for example.
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Static codes enable you to define your own renormalization scheme. You need to provide the formula in `void modify_potentials(...)` function. See [here](Strict 2D or 1D mode) for example.
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# Regularization scheme and the energy conservation in td calculations
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In publication [arXiv:1606.02225](https://arxiv.org/abs/1606.02225) it was pointed that TDBdG like equations, formally conserve energy only if all quasiparticle states are evolved, see discussion of Eqs.(25)-(26). This situation corresponds to the cubic cutoff. If the space is truncated eg. by introducing a spherical cutoff at some initial time then in general energy maybe not conserved.
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In the publication [arXiv:1606.02225](https://arxiv.org/abs/1606.02225), it was noted that TDBdG-like equations formally conserve energy only if all quasiparticle states are evolved; see the discussion of Eqs. (25)-(26). This situation corresponds to the cubic cutoff. If the space is truncated, eg, by introducing a spherical cutoff at some initial time, then in general energy may not be conserved.
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In practical applications, we observe that the energy when applied the spherical regularization scheme is conserved only with some accuracy, which is not related to the integrator accuracy. Below we provide an example of (3d calculation), where for the time interval $`te_F<170`$ we apply an external time-dependent potential (we pump energy into the system), and for $`te_F>170`$ the system evolves without any external perturbation.
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In practical applications, we observe that the energy, when applying the spherical regularization scheme, is conserved only with some accuracy, which is not related to the integrator's accuracy. Below we provide an example of (3d calculation), where for the time interval $`te_F<170`$ we apply an external time-dependent potential (we pump energy into the system), and for $`te_F>170`$ the system evolves without any external perturbation.
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It is clearly visible, that for evolution with the cubic cutoff the energy is conserved up to high accuracy, while for spherical cutoff the quality of the energy conservation is significantly lower.
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It is clearly visible that for evolution with the cubic cutoff, energy is conserved with high accuracy, whereas for the spherical cutoff, the quality of energy conservation is significantly lower.
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`wlog` files for these runs:
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* [cubic.wlog](uploads/5aff56cbbd8ee97046cc1c3d10a49867/cubic.wlog)
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* [spherical.wlog](uploads/e642fe76e9496d2305f4c462ebc61e72/spherical.wlog)
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In conclusion, we find that typically for trajectories of length $`te_F\approx1000`$ the spherical cutoff provides reasonable accuracy, while for generation of long trajectories $`te_F\gg 1000`$ it is recommended to use the cubic cutoff.
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In conclusion, we find that typically for trajectories of length $`te_F\approx1000`$ the spherical cutoff provides reasonable accuracy, while for the generation of long trajectories $`te_F\gg 1000`$ it is recommended to use the cubic cutoff.
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# Known issues
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Below we present results for the uniform unitary Fermi gas as a function of lattice spacing, while keeping the fixed volume of the box. Note that the lattice spacing defines value of the energy cut-off $`E_c\approx\frac{p_c^2}{2}=\frac{\pi^2}{2DX^2}`$. Conditions of the test are as follow:
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```
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# VOLUME: 32 x 32 x 32
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# ENERGY DENSITY FUNCTIONAL: SLDA
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# UNIFORM_TEST_MODE: Setting number of particles to be: (554.000000,554.000000)
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# Impact of the regularization scheme on the quality of static results
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Below, we present results for the uniform unitary Fermi gas as a function of gas density measured by $`k_F=(3\pi^2 n)^{1/3}`$ and scaled with respect to the cut-off momentum $`k_c=\pi/dx`$. Note that the lattice spacing defines the value of the energy cut-off $` E_c\approx\frac{k_c^2}{2}=\frac{\pi^2}{2dx^2}`$. It is a technical parameter, so the results should not depend on its choice (assuming that it is chosen from a reasonable range). We also note that typical applications utilize $`k_F dx\approx 1`$ (or $`k_F dx/\pi\approx 0.32`$).
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For the test, we used ASLDA function with the parameters provided in https://arxiv.org/abs/1008.3933, and which were adjusted in such a way to provide for the spin-symmetric case:
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```math
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\frac{E}{E_{\rm{FG}}}=0.40(1),\quad \frac{\Delta}{\varepsilon_F}=0.504(24)
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```
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The result for the total energy is:
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The graph below shows the sensitivity of the energy and the pairing gap as a function of the $`k_F/k_c`$, for the calculations with spherical cut-off. In this case, the energy is almost not sensitive to the choice of the $`k_F/\k_c`$, while the pairing gap shows some residual dependence, but at an acceptable level.
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| DX | Spherical cut-off | Cubic cut-off |
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| -----|-------------------|---------------|
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| 1.0 | 0.39761 | 0.39926 |
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| 0.8 | 0.39834 | 0.38539 |
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| 0.5 | 0.39825 | 0.36422 |
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The next graph shows the results of the same test, but with the activated cubi cutoff scheme. In this case, we observe quantitative degradation of the quality of the results. This issue needs further investigation.
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In the case of the cubic cut-off, we observe the dependence of the lattice spacing. This issue needs further investigation.
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For raw data see: [test-spherical-vs-cubic.txt](uploads/4c97d8a36df69d5d8e1c698839e2f2ed/test-spherical-vs-cubic.txt) |
